SOLUTION: If there is a goat tied to a rectangular barn on a 50 foot lead and the barn is 20 feet by 20 feet (floor), what is the maximum grazing area? If there are regions you can't find th

Question 82176: If there is a goat tied to a rectangular barn on a 50 foot lead and the barn is 20 feet by 20 feet (floor), what is the maximum grazing area? If there are regions you can't find the area of, provide as good an estimate as you can.
Assume the goat is tied to a corner outside the barn, cannot get in, and that the barn is not grazing area.

I found only half the answer and need help finding the rest please help.
Heres what I got so far-So first i drew a circle with a 50' radius with its center at the
post. Then I drew the corner of the barn that cuts off 1/4 of the area of the circle.
Area of the circle = (pi)50^2 sq ft. = 2500(pi) sq ft
Only (3/4)(2500(pi)) is available for grazing.
That area is 1875(pi) sq ft = 5890.49 sq ft
This is what the teacher siad to do:You've found a portion of the total grazing area. But, you'll need to envision the additional partial circles that the goat can reach, if he stretches his tether around the barn. And you will have a small overlap area that should be subtracted. Thanks so much for your help!!!

Answer by scott8148(6628) (Show Source): You can put this solution on YOUR website!
when the goat reaches the corners of the barn adjacent to the tied corner, it will continue to graze in a circle but the radius is reduced by the 20 foot side of the barn

the area you are trying to find is the area you have calculated plus the two 30 foot quarter-circles minus the area where the quarter-circles overlap at the corner of the barn opposite the tied corner

the area of this overlap can be found using calculus, but a good estimate is less than 100 square feet

on further reflection, you can calculate the area directly by not letting the circles overlap and using trigonometry to find the area of the "arrowhead" shape between the circles and the barn

area from 30 foot circles is 1148.65

area from "arrowhead" is
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